This article is structured as follows. In Section
 we provide a brief summary of the
conformal methods that are required in the analysis of this
article. These include: conformal geodesics, conformal Gaussian
systems, the conformal Einstein field equations and their
constraints. For a full account of the associated literature,
the reader is referred to the monograph [1]}. Section
 provides a discussion of the various
representations of spatial infinity in the Minkowski spacetime. In
particular, it contains the explicit connection between the F-gauge
and Ashtekar's representation at spatial infinity
representation. Furthermore, it also contains a discussion on how this
connection can be expressed in terms of conformal geodesics. Section
 reviews Ashtekar's definition of asymptotically
Minkowskian spacetimes at spatial infinity (AMSI) and the associated
notion of an asymptote. It further contains a discussion of the
consequences of this definition in the light of the conformal
Einstein field equations and the associated constraint equations on
timelike hypersurfaces. Subsection REF  provides a detailed
discussion of the relation between Ashtekar's asymptote and
Friedrich's cylinder viewed as intrinsic 3-manifolds. Section 
contains the main analysis in the article —namely, the construction
of a conformal Gaussian gauge system in a neighbourhood of an
asymptote. It also provides a more detailed statement of the main
theorem of this article —Theorem
REF . In addition, the article
contains several appendices. Appendix 
provides the stability theorem for solutions to a system of ordinary
differential equations and their existence times which is used in the
analysis of the solutions to the conformal geodesic equations in a
neighbourhood of the asymptote. Appendix
 provides details of the
proof of a technical lemma (Lemma
REF ) on the regularity of geometric
fields at the asymptote. Finally Appendix 
provides details on a certain class of conformal geodesics on the
Minkowski spacetime.
